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Analytical solution for the effect of radiation on flow of a magneto-micropolar fluid past a continuously moving plate with suction and blowing
Computational Materials Science 45 (2009) 423–428
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Analytical solution for the effect of radiation on flow of a magneto-micropolar fluid past a continuously moving plate with suction and blowing M.A. Seddeek a,*, S.N. Odda b, M.Y. Akl c, M.S. Abdelmeguid d a
Department of Mathematics, Faculty of Science, Qassim University, P.O. Box 237, Burieda 81999, Saudi Arabia Department of Mathematics, Faculty of Computer Science, Qassim University, P.O. Box 237, Burieda 81999, Saudi Arabia c Department of Basic Science, Faculty of Engineering (Shopra Branch), Banha University, Cairo, Egypt d Akhbar El-Yom Academy, 6 October City, Giza 12573, Egypt b
a r t i c l e
i n f o
Article history: Received 10 June 2008 Received in revised form 26 October 2008 Accepted 4 November 2008 Available online 30 December 2008 PACS: 47.11.+j 47.60.+i 47.65.+a 47.15.x
a b s t r a c t The analytical solution is presented for the effect of radiation on flow of a magneto-micropolar fluid past a continuously moving plate with suction and blowing. The governing equations for the problem are changed to dimensionless ordinary differential equations by similarity transformation. The comparison between analytical and numerical solution has been included in the analysis. The effects of radiation parameter, magnetic field parameter, Prandtl number, coupling constant parameter and the suction or blowing parameter are discussed through graphs. Graphical results illustrating interesting features of the physics of the problem are presented and discussed. Ó 2008 Elsevier B.V. All rights reserved.
Keywords: Analytical solution Magneto-micropolar Suction Blowing Shooting method Radiation
1. Introduction Eringen [4] proposed the theory of micropolar fluids, a subclass of microfluids, in which the microscopic effects arising from the local structure and micromotions of the fluid elements are taken into account. This theory can be used to analyze the behavior of exotic lubricants, polymeric fluids and liquid crystals. Eringen [5] later generalized the micropolar fluids theory to include thermal effects. The present problem finds application in MHD generators with neutral fluid seeding in the form of rigid microinclusions. Also, many industrial applications involve fluids as a working medium, and in such applications unclean fluids (i.e. clean fluid plus interspersed particles) are the rule and clean fluids an exception. A comprehensive review of the subject and application of micropolar fluid mechanics is given by Ariman et al. [6]. Ahmadi [7] obtained a similarity solution for the micropolar boundary layer flow over a semi-infinite plate. Hassanien [8], used the theory of micropolar * Corresponding author. E-mail address: [email protected] (M.A. Seddeek). 0927-0256/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2008.11.001
fluids formulated by Eringen to investigate the mixed convection boundary layer flow over a horizontal semi-infinite plate. OrtegaTorres and Rojas-Medar [9] studied the initial value problem for the equations of magneto-micropolar fluid in a time dependent domain. Seddeek [10] studied the effects of Hall and ion-slip currents on magneto-micropolar fluid and heat transfer over a non-isothermal stretching sheet with suction and blowing. Also, Seddeek [11] studied the flow of a micropolar fluid past a moving plate by the presence of magnetic field. Hossain and Takhar [12] have analyzed the effect of radiation using the Rosseland diffusion approximation, which leads to nonsimilar solutions for an optically dense viscous incompressible fluid past a heated vertical plate with uniform free-stream velocity and surface temperature. Perdikis and Raptis [13] studied the heat of a micropolar fluid by the presence of radiation. Raptis [3] used the theory of micropolar fluid to study the flow of a micropolar fluid past a moving plate in the presence of radiation. Elbashbeshy and Bazid [14] have analyzed the effect of radiation on forced convection flow of a micropolar fluid over a horizontal semi-infinite plate. Recently, Seddeek et al. [15–17] investigated effects of
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radiation on heat transfer over a stretching surface and porous media. The theory of thermomicropolar fluids includes the effects of local rotary inertia and coupled stresses, which are not present in the theory of Newtonian fluids. A fluid with suspensions, polymeric fluid, liquid crystals, and animal blood may be characterized as micropolar fluids (see Tozeren and Skalak [18] and Liu [19]). The theory of micropolar fluids is generating a lot of interest and many classical flows are being reexamined to determine the effects of fluid microstructure. The above investigations for the case of classical Newtonian fluids do not give satisfactory results (Hoyt and Fabula [20] and Vogel and Patterson [21]) if the fluid is a heterogeneous mixture such as liquid crystal, ferro liquid, or a liquid with polymer additives. These are realistic and important from a technological point of view. Hence, the aim of the present work is to study the effects of radiation parameter, magnetic field parameter, Prandtl number, coupling constant parameter and the suction or blowing parameter on heat transfer from moving plate in a steady, incompressible, micropolar fluid. We have reduced the two-dimensional continuity, momentum, angular momentum and energy equations to a system of nonlinear ordinary differential equations which are solved analytically and numerically. The effects of various parameters on the flow and heat transfer have been shown in tables and graphically. 2. Mathematical formulation Let us consider the steady, laminar, incompressible, viscous, micropolar and electrically conducting fluid flowing past a continuously moving plate with a constant velocity. The flow is assumed to be in the x-axis, which is taken along the plate and the y-axis perpendicular to it. The origin is located at the spot through which the plate is drawn in the fluid medium. A uniform strong magnetic field Bo is assumed to be applied in the y-direction and the induced magnetic field of the flow is negligible in comparison with the applied one, which corresponds to very small magnetic Reynolds number [1]. No electric field is assumed to exist and magnetic dissipation is neglected.
Continuity equation
ou ov þ ¼ 0; ox oy
ð1Þ
momentum equation
u
ou ou o2 u oN rB2o u; þ v ¼ m 2 þ k1 ox oy oy oy q1
ð2Þ
angular momentum equation
G1
o2 N ou 2N ¼ 0; oy2 oy
ð3Þ
energy equation
u
2 kf o2 T oT oT 1 oqr m ou : þv ¼ þ ox oy qcp oy2 qcp oy cp oy
ð4Þ
Here u, v are the velocity components along x, y coordinates, respectively, m is the kinematic viscosity, q is the density, N is the microrotation component, k1 ¼ qs ðk1 > 0Þ the coupling constant, s is a constant characteristic of the fluid, G1 the microrotation constant, T the fluid temperature, kf the thermal conductivity, cp is the specific heat at constant pressure and r is the electrical conductivity. By using the Rosseland diffusion approximation (Hossain et al. [2] and following Raptis [3], the radiative heat flux qr is given by
qr ¼
4r oT 4 ; 3k oy
ð5Þ
where r* is the Stefan–Boltzman constant and k* is the Rosseland mean absorption coefficient. Assuming that the temperature differences within the flow are sufficiently small such that T4 may expressed as a linear function of temperature
T 4 ffi 4T 31 T 3T 41 ;
ð6Þ
where the higher-order terms of the expansion are neglected. The boundary conditions for the present problem are
y ¼ 0 : u ¼ Bx; v ¼ vw ; N ¼ 0; T ¼ T w y ! 1 : u ! 0; N ! 0; T ! T 1
Boundary Layer
Slot
ð7Þ
In the previous equations, Tw is the temperature of the plate, T1 is the temperature of the fluid far away from the plate, B is constant and the velocity components along x, y coordinates, respectively, are
u¼
ow oy
and v ¼
ow : ox
ð8Þ
we further define the following similarity variables,
B0
rffiffiffi B y; g¼
m
y,v
pffiffiffiffiffiffi w ¼ Bmxf ðgÞ;
N¼
B3
m
!12 xgðgÞ;
h¼
T T1 Tw T1
ð9Þ
Substituting Eq. (9) into ()(2)–(4) and (7), we have 00
0
0
f 000 þ ff ðf 0 Þ2 Mf þ kg ¼ 0;
x,u
00
Bx Coordinate system for the physical model of the problem. Under the usual boundary layer approximation, the flow and heat transfer in the presence of radiation and magnetic field are governed by the following equations:
00
Gg ð2g þ f Þ ¼ 0; 1 þ R 00 h þ f h0 cf 0 h þ Ecðf 00 Þ2 ¼ 0: RPr
ð10Þ ð11Þ ð12Þ
In the above equations a prime denotes differentiation with reqc m spect to g and c is a constant, where Pr ¼ kp is the Prandtl number, 2 3kk R ¼ 16r T 3 is the radiation parameter, M ¼ qrBoB is the magnetic field 1 1 u2 k1 parameter, k ¼ m is the coupling constant parameter, Ec ¼ cp ðT wwT 1 Þ G1 B is the Eckert number and G ¼ m is the microrotation parameter.
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M.A. Seddeek et al. / Computational Materials Science 45 (2009) 423–428
The transformed boundary conditions are given by
f ð0Þ ¼ fw ; f 0 ð1Þ ¼ 0;
f 0 ð0Þ ¼ 1;
gð0Þ ¼ 0;
gð1Þ ¼ 0;
The heat flux qw at the wall is given by
hð0Þ ¼ 1
qw ¼ k
hð1Þ ¼ 0
ð13Þ
sw
Nu ¼ ð14Þ
1 xqw ¼ ðReÞ2 h0 ð0Þ: kðT w T 1 Þ
ð17Þ
3. Analytical solution
Hence the skin friction coefficient cf can be found using
cf ¼
ð16Þ
which can be used to compute the Nusselt number
The wall shear stress sw is given by
rffiffiffiffiffiffi ou uw 00 ¼ lw ¼ lw uw f ð0Þ: oy y¼0 mx
rffiffiffiffiffiffi oT uw 0 h ð0Þ ¼ kðT w T 1 Þ oy y¼0 mx
sw ¼ ðReÞ f 00 ð0Þ; qu2w 12
ð15Þ
where Re ¼ uwm x is the Reynolds number.
Eqs. (10)–(12) solved by using the method of successive approximation. First, we assume that the best zero-approximation for (10) as follows
fo ¼ fw þ að1 ebg Þ
0
Fig. 1. Effect of coupling constant parameter k on (a) the velocity profiles f (g), (b) the angular velocity profiles g(g) and (c) the temperature profiles h(g).
ð18Þ
0
Fig. 2. Effect of Magnetic parameter M on (a) the velocity profiles f (g), (b) the angular velocity profiles g(g) and (c) the temperature profiles h(g).
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M.A. Seddeek et al. / Computational Materials Science 45 (2009) 423–428
where a and b are two arbitrary constants satisfied the boundary conditions in the zero-approximation fo0 ð0Þ ¼ 1 and in the first approximation f10 ð0Þ ¼ 1. In other words ab = 1, where b is determined from
qffiffiffi 1 2f w G2k k 2 G MG Ab þ 2 þ 2M ¼ 0: b2 þ @ b4 fw b3 þ G G G 0
ð19Þ The solution of (11) is determined by using (18),
g¼
b 2
Gb 2
pffiffi2 e Gg ebg :
ð20Þ
To obtain the solution to (10) by using the method of successive approximations, the different orders are obtained from the equation 0
0
000 fiþ1 ¼ fi fi00 þ ðfi0 Þ2 þ Mf i kg ; i ¼ 0; 1; 2; . . .
Second, we suppose that the best zero-approximation of Eq. (12) is
ho ¼ edg
ð24Þ
where d is an arbitrary constant satisfied the boundary conditions and determined from
" # Rfw Pr RðEcb2 cÞPr ¼0 d dþ 1þR 1þR 2
ð25Þ
To obtain the solution to (12) by using the method of successive approximations, the different orders are obtained from the equation
h00iþ1 ¼
PrR ½fi h0 cfi0 hi þ Ecðfi00 Þ2 ; i ¼ 0; 1; 2; . . . 1þR i
ð26Þ
Integrating (26) and using the boundary conditions (13), we have
ð21Þ
Integrating (21), using (20) and the boundary conditions (13) we have,
f1 ¼ fw b þ M
!
kb2 2
Gb 2
þ1
ebg 3
b
kb 2
ðGb 2Þ
2 e
pffiffi2
g
G
G
þ c1
ð22Þ
where
" c 1 ¼ fw þ
1
b3
fw b þ M
kb2 Gb2 2
! þ1 þ
kb
#
: ðGb2 2Þ G2
ð23Þ
Fig. 3. Effect of Prandtl number Pr on the temperature profiles h(g).
0
Fig. 4. Effect of Radiation parameter R on the temperature profiles h(g).
Fig. 5. Effect of surface mass transfer parameter fw on (a) the velocity profiles f (g), (b) the angular velocity profiles g(g) and (c) the temperature profiles h(g).
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M.A. Seddeek et al. / Computational Materials Science 45 (2009) 423–428
h1 ¼
Prðfw þ aÞR dg Prðc adÞR ðbþdÞg PrEcR 2bg e þ c2 e þ e ð1 þ RÞd 4ð1 þ RÞ ð1 þ RÞðb þ dÞ2 ð27Þ
" # PrR fw þ a c ad Ec : c2 ¼ 1 þ 1þR d ðb þ dÞ2 4
ð28Þ
4. Numerical solution The resulting system of Eqs. (10)–(12) and associated boundary conditions (13) are solved by using the shooting method. First, we convert the system of Eqs. (10)–(12) into a set of coupled 1st order 0 00 0 0 equations. Letting z = [f f f g g hh ]T gives
z1
1
0
z2
3 2 3 fw f ð0Þ 7 6 7 6 0 6 f ð0Þ 7 6 1 7 7 6 7 6 00 6 f ð0Þ 7 6 n 7 7 6 7 6 7 6 7 6 zð0Þ ¼ 6 gð0Þ 7 ¼ 6 0 7: 7 6 7 6 0 6 g ð0Þ 7 6 s 7 7 6 7 6 7 6 7 6 4 hð0Þ 5 4 1 5 0 ‘ h ð0Þ 2
where
0
Second, we set this up as an IVP and use an ODE solver in Matlab to numerically integrate this system, with the initial conditions given by
1
C Bz C B z3 C B 2C B C B C B 2 B z3 C B z1 z3 þ ðz2 Þ þ Mz2 kz5 C C C B d B C B C B z5 C: B z4 C ¼ B C dg B C B 1 C B z5 C B ð2z þ z Þ 4 3 G C B C B C B C B z7 A @ z6 A @ 2 PrR ðz z c z z þ Ecðz Þ Þ z7 1 7 2 6 3 1þR
ð29Þ
ð30Þ
Third, guess n, s and ‘, and after the call to Matlab’s ODE routine, we compute the errors
E1 ¼ z2 ð1; nÞ f 0 ð1Þ;
ð31Þ
E2 ¼ z4 ð1; sÞ gð1Þ;
ð32Þ
E3 ¼ z6 ð1; ‘Þ hð1Þ:
ð33Þ
Finally, we repeat the last step until the correct value of n, s and ‘ are determined. In other words, we reach the convergence when
jEi j < tolerance ¼ 0:0001;
i ¼ 1; 2; 3:
ð34Þ
Table 1 1 1 Results of cf ðReÞ2 ; g 0 ð0Þ and NuðReÞ2 for various values of k (M = 2, Pr = 10, R = 3, fw = 0, G = 2, c = 1 and Ec = 0.02). 1
1
0
NuðReÞ2
k
b
d
cf ðReÞ2
g (0)
Analytical
Numerical
Analytical
Numerical
Analytical
Numerical
1 4
1.6399 1.3593
2.664 2.688
1.676622 1.493959
1.676305 1.494670
0.310602 0.325028
0.317949 0.321999
2.933228 2.973965
2.932841 2.974663
Table 2 1 1 Results of cf ðReÞ2 ; g 0 ð0Þ and NuðReÞ 2 for various values of M (k = 1, Pr = 10, R = 3, fw = 0, G = 2, c = 1 and Ec = 0.02). 1
1
0
NuðReÞ2
M
b
d
cf ðReÞ2
g (0)
Analytical
Numerical
Analytical
Numerical
Analytical
Numerical
1 2 3.8
1.3101 1.6399 2.112
2.691 2.664 2.614
1.358082 1.676622 2.138957
1.358217 1.676305 2.138018
0.283562 0.310602 0.339333
0.294877 0.317949 0.343713
3.033721 2.933228 2.779171
3.034414 2.932841 2.779855
Table 3 1 1 Results of cf ðReÞ2 ; g 0 ð0Þ and NuðReÞ2 for various values of Pr (k = 1, M = 2, R = 3, fw = 0, G = 2, c = 1 and Ec = 0.02). 1
1
0
NuðReÞ2
Pr
b
d
cf ðReÞ2
g (0)
Analytical
Numerical
Analytical
Numerical
Analytical
Numerical
0.71 5 10 100
1.6399 1.6399 1.6399 1.6399
0.709 1.884 2.664 8.424
1.676622 1.676622 1.676622 1.676622
1.676305 1.676305 1.676305 1.676305
0.310602 0.310602 0.310602 0.310602
0.317949 0.317949 0.317949 0.317949
0.507583 1.948456 2.933228 10.050656
0.50057 1.949998 2.932841 10.049815
Table 4 1 1 Results of cf ðReÞ2 ; g 0 ð0Þ and NuðReÞ 2 for various values of R (k = 1, M = 2, Pr = 10, fw = 0, G = 2, c = 1 and Ec = 0.02). 1
1
0
NuðReÞ2
R
b
d
cf ðReÞ2
g (0)
Analytical
Numerical
Analytical
Numerical
Analytical
Numerical
0 0.1 0.5 1 3
1.6399 1.6399 1.6399 1.6399 1.6399
0 0.927 1.776 2.175 2.664
1.676622 1.676622 1.676622 1.676622 1.676622
1.676305 1.676305 1.676305 1.676305 1.676305
0.310602 0.310602 0.310602 0.310602 0.310602
0.317949 0.317949 0.317949 0.317949 0.317949
0 0.719536 1.807747 2.317869 2.933228
0.125000 0.742685 1.813284 2.318467 2.932841
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M.A. Seddeek et al. / Computational Materials Science 45 (2009) 423–428
Table 5 1 1 Results of cf ðReÞ2 ; g 0 ð0Þ and NuðReÞ2 for various values of fw (k = 1, M = 2, Pr = 10, R = 3, G = 2, c = 1 and Ec = 0.02). fw
b
d
0.5 0 0.5
1.4142 1.6399 1.9037
1.398 2.664 5.111
1
1
0
cf ðReÞ2
NuðReÞ2
g (0)
Analytical
Numerical
Analytical
Numerical
Analytical
Numerical
1.434341 1.676622 1.940262
1.444333 1.676305 1.946052
0.292893 0.310602 0.327805
0.300334 0.317949 0.334831
1.398473 2.933228 5.528042
1.392094 2.932841 5.527730
5. Discussion of the results 0
To study the behavior of the velocity f (g), angular velocity g(g) and temperature profiles h(g), curves are drawn for various values of the parameters that describe the flow. Fig. 1 display results for the velocity, angular velocity and tem0 perature distribution, respectively. It is seen that f (g) decreases with increasing the coupling constant parameter k, but g(g) and h(g) increase with increasing the parameter k. Also, it is clearly shown that the velocity and the angular velocity decreases with increasing the magnetic field parameter M; the Lorentz force, which opposes the flow, also increases and leads to enchanted deceleration of the flow. This conclusion meets the logic that the magnetic field exerts a retarding force on the free-convection flow. Fig. 2a and b describes the behavior of the velocity and angular velocity with changes in the values of the magnetic field parameter M. The effects of the magnetic field parameter M on the heat transfer are shown in Fig. 2c. It is observed that the temperature increases when M parameter increase. Fig. 3 shows that the prandtl number Pr has no effect on the 0 velocity f (g) and the angular velocity g(g) while the temperature distribution h(g) decreases with increasing the parameter Pr. Fig. 4 represents the effect of radiation parameter R on the temper0 ature distribution h(g) while no effect appears in the velocity f (g) and the angular velocity g(g). As the radiation parameter R increases, the temperature distribution h(g) decreases. The effect of surface mass transfer fw on the dimensionless velocity, angular velocity and temperature distributions is dis0 played in Fig. 5. The effect of suction is to decrease f (g), g(g) and 0 h(g), or that of blowing is to increase f (g), g(g) and h(g) Tables 1–5 exhibit the behavior of the skin friction coefficient at 1 00 0 the wall cf ðReÞ2 (or f (0)), wall couple stress g (0) and the local Nus1 0 selt number NuðReÞ2 (or h (0)) for various values of k, M, Pr, R and fw. The skin friction and wall couple stress increase in the presence of magnetic field and suction at the surface but they decrease in the presence of blowing at the surface. In the presence of coupling constant parameter, the wall couple stress increases but the skin friction decreases. There are no effects for the Prandtl number and radiation on the skin friction and wall couple stress. The local Nusselt number increases in the presence of coupling constant parameter, Prandtl number, radiation and suction at the surface but it decreases in the presence of magnetic field and blowing at the surface. The agreement between analytical and numerical solutions is excellent. 6. Conclusions This paper studied the effects of radiation on flow of a magnetomicropolar fluid past a continuously moving plate with suction and
blowing. The governing fundamental equations are approximated by a system of nonlinear ordinary differential equations by similarity transformation are solved analytically and numerically, the shear stress, couple stress and Nusselt number as well as the details of velocity and temperature fields are presented for various values of parameters for the problem, e.g. magnetic field (M), Prandtl number (Pr), radiation (R), surface mass transfer (fw) and coupling constant (k) parameters. 0 The numerical results indicate that, the velocity f (g) and angular velocity g(g) decrease with increasing M and fw but Pr and R not 0 affected on them, while f (g) decreases and g(g) increases with increasing k. The temperature distribution h(g) increases with increasing k and M, but h(g) decreases with increasing Pr, R and fw. The skin friction and wall couple stress increase with increasing M and fw but Pr and R not affected on them, while the skin friction decreases and wall couple stress increases with increasing k. The local Nusselt number increases with increasing k, Pr, R and fw, and decreases with increasing M. Finally, the agreement between analytical and numerical solutions is excellent. Acknowledgement The authors are grateful to the reviewers for their useful comments. References [1] V.M. Soundalgekar, M.R. Ratia and I. Pop, Magneto Hydrodynamic, second ed., vol. 65, Riga (USSK), 1982. [2] M.A. Hossain, M.A. Alim, D.A.S. Rees, Int. J. Heat Mass Transfer 42 (1999) 181. [3] A. Raptis, Int. J. Heat Mass Transfer 41 (1998) 2865. [4] A.C. Eringen, Int. J. Eng. Sci. 2 (1964) 205. [5] A.C. Eringen, J. Math. Anal. Appl. 38 (1972) 480. [6] G. Ariman, M.A. Turk, N.D. Sylvester, Int. J. Eng. Sci. 11 (1973) 905. [7] G. Ahmadi, Int. J. Eng. Sci. 14 (1976) 639. [8] I.A. Hassanien, J. Fluid Eng. 118 (1996) 833. [9] E. Ortega-Torres, M. Rajas-Medar, Mat. Contemp. 15 (1998) 295. [10] M.A. Seddeek, Proc. R. Soc. London A 457 (2001) 3039. [11] M.A. Seddeek, Phys. Lett. A 306 (2003) 255. [12] M.A. Hossain, H.S. Takhar, Int. J. Heat Mass Transfer 31 (1996) 243. [13] C. Perdikis, A. Raptis, Heat Mass Transfer 31 (1996) 381. [14] E.M.A. Elbashbeshy, M.A.A. Bazid, Can. J. Phys. 78 (2000) 907. [15] M.A. Seddeek, M.S. Abdelmeguid, Phys. Lett. A 348 (2006) 172–179. [16] M.A. Seddeek, J. Porous Media 10 (1) (2007) 99–108. [17] M.A. Seddeek, A.A. Darwish, M.S. Abdelmeguid, Commun. Nonlinear Sci. Numer. Simul. 12 (2) (2007) 195–213. [18] A. Tozeren, R. Skalak, Int. J. Eng. Sci. 15 (1977) 511. [19] C.Y. Liu, Int. J. Eng. Sci. 8 (1970) 457–466. [20] J.W. Hoyt, A.G.U.S. Fabula, Naval Ordinance Test Station Report, 1964. [21] W.M. Vogel, A.M. Patterson, Pacific Naval Lab. of the Defense Research Board of Canada Report 64 (1964) 2.
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Analytical solution for the effect of radiation on flow of a magneto-micropolar fluid past a continuously moving plate with suction and blowing M.A. Seddeek a,*, S.N. Odda b, M.Y. Akl c, M.S. Abdelmeguid d a
Department of Mathematics, Faculty of Science, Qassim University, P.O. Box 237, Burieda 81999, Saudi Arabia Department of Mathematics, Faculty of Computer Science, Qassim University, P.O. Box 237, Burieda 81999, Saudi Arabia c Department of Basic Science, Faculty of Engineering (Shopra Branch), Banha University, Cairo, Egypt d Akhbar El-Yom Academy, 6 October City, Giza 12573, Egypt b
a r t i c l e
i n f o
Article history: Received 10 June 2008 Received in revised form 26 October 2008 Accepted 4 November 2008 Available online 30 December 2008 PACS: 47.11.+j 47.60.+i 47.65.+a 47.15.x
a b s t r a c t The analytical solution is presented for the effect of radiation on flow of a magneto-micropolar fluid past a continuously moving plate with suction and blowing. The governing equations for the problem are changed to dimensionless ordinary differential equations by similarity transformation. The comparison between analytical and numerical solution has been included in the analysis. The effects of radiation parameter, magnetic field parameter, Prandtl number, coupling constant parameter and the suction or blowing parameter are discussed through graphs. Graphical results illustrating interesting features of the physics of the problem are presented and discussed. Ó 2008 Elsevier B.V. All rights reserved.
Keywords: Analytical solution Magneto-micropolar Suction Blowing Shooting method Radiation
1. Introduction Eringen [4] proposed the theory of micropolar fluids, a subclass of microfluids, in which the microscopic effects arising from the local structure and micromotions of the fluid elements are taken into account. This theory can be used to analyze the behavior of exotic lubricants, polymeric fluids and liquid crystals. Eringen [5] later generalized the micropolar fluids theory to include thermal effects. The present problem finds application in MHD generators with neutral fluid seeding in the form of rigid microinclusions. Also, many industrial applications involve fluids as a working medium, and in such applications unclean fluids (i.e. clean fluid plus interspersed particles) are the rule and clean fluids an exception. A comprehensive review of the subject and application of micropolar fluid mechanics is given by Ariman et al. [6]. Ahmadi [7] obtained a similarity solution for the micropolar boundary layer flow over a semi-infinite plate. Hassanien [8], used the theory of micropolar * Corresponding author. E-mail address: [email protected] (M.A. Seddeek). 0927-0256/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2008.11.001
fluids formulated by Eringen to investigate the mixed convection boundary layer flow over a horizontal semi-infinite plate. OrtegaTorres and Rojas-Medar [9] studied the initial value problem for the equations of magneto-micropolar fluid in a time dependent domain. Seddeek [10] studied the effects of Hall and ion-slip currents on magneto-micropolar fluid and heat transfer over a non-isothermal stretching sheet with suction and blowing. Also, Seddeek [11] studied the flow of a micropolar fluid past a moving plate by the presence of magnetic field. Hossain and Takhar [12] have analyzed the effect of radiation using the Rosseland diffusion approximation, which leads to nonsimilar solutions for an optically dense viscous incompressible fluid past a heated vertical plate with uniform free-stream velocity and surface temperature. Perdikis and Raptis [13] studied the heat of a micropolar fluid by the presence of radiation. Raptis [3] used the theory of micropolar fluid to study the flow of a micropolar fluid past a moving plate in the presence of radiation. Elbashbeshy and Bazid [14] have analyzed the effect of radiation on forced convection flow of a micropolar fluid over a horizontal semi-infinite plate. Recently, Seddeek et al. [15–17] investigated effects of
424
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radiation on heat transfer over a stretching surface and porous media. The theory of thermomicropolar fluids includes the effects of local rotary inertia and coupled stresses, which are not present in the theory of Newtonian fluids. A fluid with suspensions, polymeric fluid, liquid crystals, and animal blood may be characterized as micropolar fluids (see Tozeren and Skalak [18] and Liu [19]). The theory of micropolar fluids is generating a lot of interest and many classical flows are being reexamined to determine the effects of fluid microstructure. The above investigations for the case of classical Newtonian fluids do not give satisfactory results (Hoyt and Fabula [20] and Vogel and Patterson [21]) if the fluid is a heterogeneous mixture such as liquid crystal, ferro liquid, or a liquid with polymer additives. These are realistic and important from a technological point of view. Hence, the aim of the present work is to study the effects of radiation parameter, magnetic field parameter, Prandtl number, coupling constant parameter and the suction or blowing parameter on heat transfer from moving plate in a steady, incompressible, micropolar fluid. We have reduced the two-dimensional continuity, momentum, angular momentum and energy equations to a system of nonlinear ordinary differential equations which are solved analytically and numerically. The effects of various parameters on the flow and heat transfer have been shown in tables and graphically. 2. Mathematical formulation Let us consider the steady, laminar, incompressible, viscous, micropolar and electrically conducting fluid flowing past a continuously moving plate with a constant velocity. The flow is assumed to be in the x-axis, which is taken along the plate and the y-axis perpendicular to it. The origin is located at the spot through which the plate is drawn in the fluid medium. A uniform strong magnetic field Bo is assumed to be applied in the y-direction and the induced magnetic field of the flow is negligible in comparison with the applied one, which corresponds to very small magnetic Reynolds number [1]. No electric field is assumed to exist and magnetic dissipation is neglected.
Continuity equation
ou ov þ ¼ 0; ox oy
ð1Þ
momentum equation
u
ou ou o2 u oN rB2o u; þ v ¼ m 2 þ k1 ox oy oy oy q1
ð2Þ
angular momentum equation
G1
o2 N ou 2N ¼ 0; oy2 oy
ð3Þ
energy equation
u
2 kf o2 T oT oT 1 oqr m ou : þv ¼ þ ox oy qcp oy2 qcp oy cp oy
ð4Þ
Here u, v are the velocity components along x, y coordinates, respectively, m is the kinematic viscosity, q is the density, N is the microrotation component, k1 ¼ qs ðk1 > 0Þ the coupling constant, s is a constant characteristic of the fluid, G1 the microrotation constant, T the fluid temperature, kf the thermal conductivity, cp is the specific heat at constant pressure and r is the electrical conductivity. By using the Rosseland diffusion approximation (Hossain et al. [2] and following Raptis [3], the radiative heat flux qr is given by
qr ¼
4r oT 4 ; 3k oy
ð5Þ
where r* is the Stefan–Boltzman constant and k* is the Rosseland mean absorption coefficient. Assuming that the temperature differences within the flow are sufficiently small such that T4 may expressed as a linear function of temperature
T 4 ffi 4T 31 T 3T 41 ;
ð6Þ
where the higher-order terms of the expansion are neglected. The boundary conditions for the present problem are
y ¼ 0 : u ¼ Bx; v ¼ vw ; N ¼ 0; T ¼ T w y ! 1 : u ! 0; N ! 0; T ! T 1
Boundary Layer
Slot
ð7Þ
In the previous equations, Tw is the temperature of the plate, T1 is the temperature of the fluid far away from the plate, B is constant and the velocity components along x, y coordinates, respectively, are
u¼
ow oy
and v ¼
ow : ox
ð8Þ
we further define the following similarity variables,
B0
rffiffiffi B y; g¼
m
y,v
pffiffiffiffiffiffi w ¼ Bmxf ðgÞ;
N¼
B3
m
!12 xgðgÞ;
h¼
T T1 Tw T1
ð9Þ
Substituting Eq. (9) into ()(2)–(4) and (7), we have 00
0
0
f 000 þ ff ðf 0 Þ2 Mf þ kg ¼ 0;
x,u
00
Bx Coordinate system for the physical model of the problem. Under the usual boundary layer approximation, the flow and heat transfer in the presence of radiation and magnetic field are governed by the following equations:
00
Gg ð2g þ f Þ ¼ 0; 1 þ R 00 h þ f h0 cf 0 h þ Ecðf 00 Þ2 ¼ 0: RPr
ð10Þ ð11Þ ð12Þ
In the above equations a prime denotes differentiation with reqc m spect to g and c is a constant, where Pr ¼ kp is the Prandtl number, 2 3kk R ¼ 16r T 3 is the radiation parameter, M ¼ qrBoB is the magnetic field 1 1 u2 k1 parameter, k ¼ m is the coupling constant parameter, Ec ¼ cp ðT wwT 1 Þ G1 B is the Eckert number and G ¼ m is the microrotation parameter.
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The transformed boundary conditions are given by
f ð0Þ ¼ fw ; f 0 ð1Þ ¼ 0;
f 0 ð0Þ ¼ 1;
gð0Þ ¼ 0;
gð1Þ ¼ 0;
The heat flux qw at the wall is given by
hð0Þ ¼ 1
qw ¼ k
hð1Þ ¼ 0
ð13Þ
sw
Nu ¼ ð14Þ
1 xqw ¼ ðReÞ2 h0 ð0Þ: kðT w T 1 Þ
ð17Þ
3. Analytical solution
Hence the skin friction coefficient cf can be found using
cf ¼
ð16Þ
which can be used to compute the Nusselt number
The wall shear stress sw is given by
rffiffiffiffiffiffi ou uw 00 ¼ lw ¼ lw uw f ð0Þ: oy y¼0 mx
rffiffiffiffiffiffi oT uw 0 h ð0Þ ¼ kðT w T 1 Þ oy y¼0 mx
sw ¼ ðReÞ f 00 ð0Þ; qu2w 12
ð15Þ
where Re ¼ uwm x is the Reynolds number.
Eqs. (10)–(12) solved by using the method of successive approximation. First, we assume that the best zero-approximation for (10) as follows
fo ¼ fw þ að1 ebg Þ
0
Fig. 1. Effect of coupling constant parameter k on (a) the velocity profiles f (g), (b) the angular velocity profiles g(g) and (c) the temperature profiles h(g).
ð18Þ
0
Fig. 2. Effect of Magnetic parameter M on (a) the velocity profiles f (g), (b) the angular velocity profiles g(g) and (c) the temperature profiles h(g).
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M.A. Seddeek et al. / Computational Materials Science 45 (2009) 423–428
where a and b are two arbitrary constants satisfied the boundary conditions in the zero-approximation fo0 ð0Þ ¼ 1 and in the first approximation f10 ð0Þ ¼ 1. In other words ab = 1, where b is determined from
qffiffiffi 1 2f w G2k k 2 G MG Ab þ 2 þ 2M ¼ 0: b2 þ @ b4 fw b3 þ G G G 0
ð19Þ The solution of (11) is determined by using (18),
g¼
b 2
Gb 2
pffiffi2 e Gg ebg :
ð20Þ
To obtain the solution to (10) by using the method of successive approximations, the different orders are obtained from the equation 0
0
000 fiþ1 ¼ fi fi00 þ ðfi0 Þ2 þ Mf i kg ; i ¼ 0; 1; 2; . . .
Second, we suppose that the best zero-approximation of Eq. (12) is
ho ¼ edg
ð24Þ
where d is an arbitrary constant satisfied the boundary conditions and determined from
" # Rfw Pr RðEcb2 cÞPr ¼0 d dþ 1þR 1þR 2
ð25Þ
To obtain the solution to (12) by using the method of successive approximations, the different orders are obtained from the equation
h00iþ1 ¼
PrR ½fi h0 cfi0 hi þ Ecðfi00 Þ2 ; i ¼ 0; 1; 2; . . . 1þR i
ð26Þ
Integrating (26) and using the boundary conditions (13), we have
ð21Þ
Integrating (21), using (20) and the boundary conditions (13) we have,
f1 ¼ fw b þ M
!
kb2 2
Gb 2
þ1
ebg 3
b
kb 2
ðGb 2Þ
2 e
pffiffi2
g
G
G
þ c1
ð22Þ
where
" c 1 ¼ fw þ
1
b3
fw b þ M
kb2 Gb2 2
! þ1 þ
kb
#
: ðGb2 2Þ G2
ð23Þ
Fig. 3. Effect of Prandtl number Pr on the temperature profiles h(g).
0
Fig. 4. Effect of Radiation parameter R on the temperature profiles h(g).
Fig. 5. Effect of surface mass transfer parameter fw on (a) the velocity profiles f (g), (b) the angular velocity profiles g(g) and (c) the temperature profiles h(g).
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M.A. Seddeek et al. / Computational Materials Science 45 (2009) 423–428
h1 ¼
Prðfw þ aÞR dg Prðc adÞR ðbþdÞg PrEcR 2bg e þ c2 e þ e ð1 þ RÞd 4ð1 þ RÞ ð1 þ RÞðb þ dÞ2 ð27Þ
" # PrR fw þ a c ad Ec : c2 ¼ 1 þ 1þR d ðb þ dÞ2 4
ð28Þ
4. Numerical solution The resulting system of Eqs. (10)–(12) and associated boundary conditions (13) are solved by using the shooting method. First, we convert the system of Eqs. (10)–(12) into a set of coupled 1st order 0 00 0 0 equations. Letting z = [f f f g g hh ]T gives
z1
1
0
z2
3 2 3 fw f ð0Þ 7 6 7 6 0 6 f ð0Þ 7 6 1 7 7 6 7 6 00 6 f ð0Þ 7 6 n 7 7 6 7 6 7 6 7 6 zð0Þ ¼ 6 gð0Þ 7 ¼ 6 0 7: 7 6 7 6 0 6 g ð0Þ 7 6 s 7 7 6 7 6 7 6 7 6 4 hð0Þ 5 4 1 5 0 ‘ h ð0Þ 2
where
0
Second, we set this up as an IVP and use an ODE solver in Matlab to numerically integrate this system, with the initial conditions given by
1
C Bz C B z3 C B 2C B C B C B 2 B z3 C B z1 z3 þ ðz2 Þ þ Mz2 kz5 C C C B d B C B C B z5 C: B z4 C ¼ B C dg B C B 1 C B z5 C B ð2z þ z Þ 4 3 G C B C B C B C B z7 A @ z6 A @ 2 PrR ðz z c z z þ Ecðz Þ Þ z7 1 7 2 6 3 1þR
ð29Þ
ð30Þ
Third, guess n, s and ‘, and after the call to Matlab’s ODE routine, we compute the errors
E1 ¼ z2 ð1; nÞ f 0 ð1Þ;
ð31Þ
E2 ¼ z4 ð1; sÞ gð1Þ;
ð32Þ
E3 ¼ z6 ð1; ‘Þ hð1Þ:
ð33Þ
Finally, we repeat the last step until the correct value of n, s and ‘ are determined. In other words, we reach the convergence when
jEi j < tolerance ¼ 0:0001;
i ¼ 1; 2; 3:
ð34Þ
Table 1 1 1 Results of cf ðReÞ2 ; g 0 ð0Þ and NuðReÞ2 for various values of k (M = 2, Pr = 10, R = 3, fw = 0, G = 2, c = 1 and Ec = 0.02). 1
1
0
NuðReÞ2
k
b
d
cf ðReÞ2
g (0)
Analytical
Numerical
Analytical
Numerical
Analytical
Numerical
1 4
1.6399 1.3593
2.664 2.688
1.676622 1.493959
1.676305 1.494670
0.310602 0.325028
0.317949 0.321999
2.933228 2.973965
2.932841 2.974663
Table 2 1 1 Results of cf ðReÞ2 ; g 0 ð0Þ and NuðReÞ 2 for various values of M (k = 1, Pr = 10, R = 3, fw = 0, G = 2, c = 1 and Ec = 0.02). 1
1
0
NuðReÞ2
M
b
d
cf ðReÞ2
g (0)
Analytical
Numerical
Analytical
Numerical
Analytical
Numerical
1 2 3.8
1.3101 1.6399 2.112
2.691 2.664 2.614
1.358082 1.676622 2.138957
1.358217 1.676305 2.138018
0.283562 0.310602 0.339333
0.294877 0.317949 0.343713
3.033721 2.933228 2.779171
3.034414 2.932841 2.779855
Table 3 1 1 Results of cf ðReÞ2 ; g 0 ð0Þ and NuðReÞ2 for various values of Pr (k = 1, M = 2, R = 3, fw = 0, G = 2, c = 1 and Ec = 0.02). 1
1
0
NuðReÞ2
Pr
b
d
cf ðReÞ2
g (0)
Analytical
Numerical
Analytical
Numerical
Analytical
Numerical
0.71 5 10 100
1.6399 1.6399 1.6399 1.6399
0.709 1.884 2.664 8.424
1.676622 1.676622 1.676622 1.676622
1.676305 1.676305 1.676305 1.676305
0.310602 0.310602 0.310602 0.310602
0.317949 0.317949 0.317949 0.317949
0.507583 1.948456 2.933228 10.050656
0.50057 1.949998 2.932841 10.049815
Table 4 1 1 Results of cf ðReÞ2 ; g 0 ð0Þ and NuðReÞ 2 for various values of R (k = 1, M = 2, Pr = 10, fw = 0, G = 2, c = 1 and Ec = 0.02). 1
1
0
NuðReÞ2
R
b
d
cf ðReÞ2
g (0)
Analytical
Numerical
Analytical
Numerical
Analytical
Numerical
0 0.1 0.5 1 3
1.6399 1.6399 1.6399 1.6399 1.6399
0 0.927 1.776 2.175 2.664
1.676622 1.676622 1.676622 1.676622 1.676622
1.676305 1.676305 1.676305 1.676305 1.676305
0.310602 0.310602 0.310602 0.310602 0.310602
0.317949 0.317949 0.317949 0.317949 0.317949
0 0.719536 1.807747 2.317869 2.933228
0.125000 0.742685 1.813284 2.318467 2.932841
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M.A. Seddeek et al. / Computational Materials Science 45 (2009) 423–428
Table 5 1 1 Results of cf ðReÞ2 ; g 0 ð0Þ and NuðReÞ2 for various values of fw (k = 1, M = 2, Pr = 10, R = 3, G = 2, c = 1 and Ec = 0.02). fw
b
d
0.5 0 0.5
1.4142 1.6399 1.9037
1.398 2.664 5.111
1
1
0
cf ðReÞ2
NuðReÞ2
g (0)
Analytical
Numerical
Analytical
Numerical
Analytical
Numerical
1.434341 1.676622 1.940262
1.444333 1.676305 1.946052
0.292893 0.310602 0.327805
0.300334 0.317949 0.334831
1.398473 2.933228 5.528042
1.392094 2.932841 5.527730
5. Discussion of the results 0
To study the behavior of the velocity f (g), angular velocity g(g) and temperature profiles h(g), curves are drawn for various values of the parameters that describe the flow. Fig. 1 display results for the velocity, angular velocity and tem0 perature distribution, respectively. It is seen that f (g) decreases with increasing the coupling constant parameter k, but g(g) and h(g) increase with increasing the parameter k. Also, it is clearly shown that the velocity and the angular velocity decreases with increasing the magnetic field parameter M; the Lorentz force, which opposes the flow, also increases and leads to enchanted deceleration of the flow. This conclusion meets the logic that the magnetic field exerts a retarding force on the free-convection flow. Fig. 2a and b describes the behavior of the velocity and angular velocity with changes in the values of the magnetic field parameter M. The effects of the magnetic field parameter M on the heat transfer are shown in Fig. 2c. It is observed that the temperature increases when M parameter increase. Fig. 3 shows that the prandtl number Pr has no effect on the 0 velocity f (g) and the angular velocity g(g) while the temperature distribution h(g) decreases with increasing the parameter Pr. Fig. 4 represents the effect of radiation parameter R on the temper0 ature distribution h(g) while no effect appears in the velocity f (g) and the angular velocity g(g). As the radiation parameter R increases, the temperature distribution h(g) decreases. The effect of surface mass transfer fw on the dimensionless velocity, angular velocity and temperature distributions is dis0 played in Fig. 5. The effect of suction is to decrease f (g), g(g) and 0 h(g), or that of blowing is to increase f (g), g(g) and h(g) Tables 1–5 exhibit the behavior of the skin friction coefficient at 1 00 0 the wall cf ðReÞ2 (or f (0)), wall couple stress g (0) and the local Nus1 0 selt number NuðReÞ2 (or h (0)) for various values of k, M, Pr, R and fw. The skin friction and wall couple stress increase in the presence of magnetic field and suction at the surface but they decrease in the presence of blowing at the surface. In the presence of coupling constant parameter, the wall couple stress increases but the skin friction decreases. There are no effects for the Prandtl number and radiation on the skin friction and wall couple stress. The local Nusselt number increases in the presence of coupling constant parameter, Prandtl number, radiation and suction at the surface but it decreases in the presence of magnetic field and blowing at the surface. The agreement between analytical and numerical solutions is excellent. 6. Conclusions This paper studied the effects of radiation on flow of a magnetomicropolar fluid past a continuously moving plate with suction and
blowing. The governing fundamental equations are approximated by a system of nonlinear ordinary differential equations by similarity transformation are solved analytically and numerically, the shear stress, couple stress and Nusselt number as well as the details of velocity and temperature fields are presented for various values of parameters for the problem, e.g. magnetic field (M), Prandtl number (Pr), radiation (R), surface mass transfer (fw) and coupling constant (k) parameters. 0 The numerical results indicate that, the velocity f (g) and angular velocity g(g) decrease with increasing M and fw but Pr and R not 0 affected on them, while f (g) decreases and g(g) increases with increasing k. The temperature distribution h(g) increases with increasing k and M, but h(g) decreases with increasing Pr, R and fw. The skin friction and wall couple stress increase with increasing M and fw but Pr and R not affected on them, while the skin friction decreases and wall couple stress increases with increasing k. The local Nusselt number increases with increasing k, Pr, R and fw, and decreases with increasing M. Finally, the agreement between analytical and numerical solutions is excellent. Acknowledgement The authors are grateful to the reviewers for their useful comments. References [1] V.M. Soundalgekar, M.R. Ratia and I. Pop, Magneto Hydrodynamic, second ed., vol. 65, Riga (USSK), 1982. [2] M.A. Hossain, M.A. Alim, D.A.S. Rees, Int. J. Heat Mass Transfer 42 (1999) 181. [3] A. Raptis, Int. J. Heat Mass Transfer 41 (1998) 2865. [4] A.C. Eringen, Int. J. Eng. Sci. 2 (1964) 205. [5] A.C. Eringen, J. Math. Anal. Appl. 38 (1972) 480. [6] G. Ariman, M.A. Turk, N.D. Sylvester, Int. J. Eng. Sci. 11 (1973) 905. [7] G. Ahmadi, Int. J. Eng. Sci. 14 (1976) 639. [8] I.A. Hassanien, J. Fluid Eng. 118 (1996) 833. [9] E. Ortega-Torres, M. Rajas-Medar, Mat. Contemp. 15 (1998) 295. [10] M.A. Seddeek, Proc. R. Soc. London A 457 (2001) 3039. [11] M.A. Seddeek, Phys. Lett. A 306 (2003) 255. [12] M.A. Hossain, H.S. Takhar, Int. J. Heat Mass Transfer 31 (1996) 243. [13] C. Perdikis, A. Raptis, Heat Mass Transfer 31 (1996) 381. [14] E.M.A. Elbashbeshy, M.A.A. Bazid, Can. J. Phys. 78 (2000) 907. [15] M.A. Seddeek, M.S. Abdelmeguid, Phys. Lett. A 348 (2006) 172–179. [16] M.A. Seddeek, J. Porous Media 10 (1) (2007) 99–108. [17] M.A. Seddeek, A.A. Darwish, M.S. Abdelmeguid, Commun. Nonlinear Sci. Numer. Simul. 12 (2) (2007) 195–213. [18] A. Tozeren, R. Skalak, Int. J. Eng. Sci. 15 (1977) 511. [19] C.Y. Liu, Int. J. Eng. Sci. 8 (1970) 457–466. [20] J.W. Hoyt, A.G.U.S. Fabula, Naval Ordinance Test Station Report, 1964. [21] W.M. Vogel, A.M. Patterson, Pacific Naval Lab. of the Defense Research Board of Canada Report 64 (1964) 2.